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\section*{derived category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Properties}{Equivalent characterizations and constructions}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{AsLOcalizationAtNullSystem}{In terms of localization at a null system}\dotfill \pageref*{AsLOcalizationAtNullSystem} \linebreak
\noindent\hyperlink{InTermsOfResolutions}{In terms of injective and projective resolutions}\dotfill \pageref*{InTermsOfResolutions} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{derived (infinity,1)-category} or \emph{derived category} of an [[abelian category]] $\mathcal{A}$ is the setting for [[homological algebra]] in $\mathcal{A}$: the [[(infinity,1)-categorical localization]] of the [[category of chain complexes]] in $\mathcal{A}$ at the class of [[quasi-isomorphisms]]. The derived category is a fundamental example of a [[stable (infinity,1)-category]]. By the [[stable Dold-Kan correspondence]], it may be viewed as a linearization of the [[stable (infinity,1)-category of spectra]].

The derived (infinity,1)-category is presented by various [[dg-model structures]] on the [[category of chain complexes]], as described at [[model structures on chain complexes]]. As such it has also a natural incarnation as a [[pretriangulated dg-category]], which might be called the \emph{derived dg-category}.

Like any [[stable (infinity,1)-category]], the [[homotopy category of an (infinity,1)-category|homotopy category]] of the derived (infinity,1)-category admits a canonical [[triangulated category]] structure. Often in the literature, the term \emph{derived category} refers to the [[homotopy category of an (infinity,1)-category|homotopy category]], viewed only as a [[triangulated category]]. The loss of information can often be problematic, but for many purposes is not important.

In what follows, we will describe only the homotopy category. See [[(infinity,1)-category of chain complexes]] for the full [[(infinity,1)-category]].

Associated to $\mathcal{A}$ is

\begin{itemize}%
\item the [[category of chain complexes]] $Ch_\bullet(\mathcal{A})$ in $\mathcal{A}$ which is naturally a [[homotopical category]];


\item the ``[[homotopy category of chain complexes]]'' $K(\mathcal{A})$;


\item the [[stable ∞-category]] $K_\infty(\mathcal{A})$ of [[chain complexes]] in $C$.



\end{itemize}
The \emph{derived category} $D(C)$ of $C$ is equivalently

\begin{itemize}%
\item the [[homotopy category|1-categorical homotopy category]] of $Ch_\bullet(\mathcal{A})$ with respect to the [[quasi-isomorphisms]];


\item the [[homotopy category of an (infinity,1)-category|(∞,1)-categorical homotopy category]] of $K_\infty(\mathcal{A})$.



\end{itemize}
In either case, this means that under the canonical [[localization]] functor

\begin{displaymath}
Q : Ch_\bullet(\mathcal{A}) \to D(\mathcal{A})
\end{displaymath}
the [[quasi-isomorphisms]] of [[chain complexes]] become true [[isomorphisms]] and that $D(\mathcal{A})$ is [[universal property|universal]] with respect to this property.

Hence the derived category is an approximation to the full [[simplicial localization]] of $K(\mathcal{A})$. It is or can be equipped with several further [[properties]] and [[structure]] that give a more accurate approximation. Notably every derived category is a \emph{[[triangulated category]]}, which is a way of remembering the [[suspension]] and de-suspension operations on its objects -- the [[suspension of chain complexes]] -- hence its ``[[stable (infinity,1)-category|stability]]''.

\hypertarget{history}{}\subsection*{{History}}\label{history}

Derived categories were introduced by [[Jean-Louis Verdier]] in his thesis under the supervision of [[Alexandre Grothendieck]]. It was originally used to extend [[Serre duality]] to a relative context. See [[Robin Hartshorne|Hartshorne]]`s lecture notes ``Residues and duality''.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $\mathcal{A}$ be an [[abelian category]] and $K(\mathcal{A})$ its [[category of chain complexes]] modulo [[chain homotopy]] (the ``[[homotopy category of chain complexes]]'').

Equip $K(\mathcal{A})$ with the structure of a [[homotopical category]] by declaring the [[weak equivalences]] to be the \textbf{quasi-isomorphisms}: those morphisms $f : V \to W$ which induce [[isomorphisms]] in [[homology]], $H(f) : H(V) \stackrel{\simeq}{\to} H(W)$.

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{derived category} $D(\mathcal{A})$ is the [[homotopy category]] of $K(\mathcal{A})$ with respect to these weak equivalences.

\end{defn}
Analogously, for $K^{+,-,b}(\mathcal{A})$ denoting the [[full subcategory]] on the chain complexes bounded above, bounded below, or bounded, respectively (see at \emph{[[category of chain complexes]]}), one writes

\begin{displaymath}
D^{+,-,b}(\mathcal{A}) \hookrightarrow D(\mathcal{A})
\end{displaymath}
for the correspponding full subcategory of the derived category.

\hypertarget{Properties}{}\subsection*{{Equivalent characterizations and constructions}}\label{Properties}

There are various ways to construct or express the derived category more explicitly in terms of various special objects or morphisms in the [[category of chain complexes]].

\begin{itemize}%
\item \emph{\hyperlink{AsLOcalizationAtNullSystem}{In terms of localization at a null system}}


\item \emph{\hyperlink{InTermsOfResolutions}{In terms of resolutions}}.



\end{itemize}
\hypertarget{AsLOcalizationAtNullSystem}{}\subsubsection*{{In terms of localization at a null system}}\label{AsLOcalizationAtNullSystem}

The ``[[homotopy category of chain complexes]]'' $K(\mathcal{A})$ is already a [[triangulated category]]. The derived category can be obtained as the construction of a [[homotopy category]] of a [[triangulated category]] with respect to a [[null system]].

\begin{defn}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
N(\mathcal{A}) \subset K(\mathcal{A})
\end{displaymath}
and analogously

\begin{displaymath}
N^{+,-n,b}(\mathcal{A}) \subset K^{+,-,b}(\mathcal{A})
\end{displaymath}
be the [[full subcategory]] of $K(C)$ or on $K^{+,-,b}$, respectively, on those [[chain complexes]] $V$ whose [[chain homology]] vanishes in every degree, $H_\bullet(V) = 0$.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
A [[chain map]] $f_\bullet : V_\bullet \to W_\bullet$ is a [[quasi-isomorphism]] precisely there exists a [[distinguished triangle]] in $K(\mathcal{A})$ of the form

\begin{displaymath}
V \stackrel{f}{\to} W \to cone(f)
\end{displaymath}
with the [[mapping cone]] $cone(f) \in N(\mathcal{C})$.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
The derived category is equivalently the localization of $K(\mathcal{A})$ at the [[null system]] $N(\mathcal{A})$.

\begin{displaymath}
D(\mathcal{A}) \simeq K(\mathcal{A})/N(\mathcal{A})
  \,.
\end{displaymath}
\end{prop}
This perspective is discussed in (\hyperlink{KashiwaraSchapira}{Kashiwara-Schapira, section 13}) and (\hyperlink{Schapira}{Schapira, section 6.2, 72}).

\hypertarget{InTermsOfResolutions}{}\subsubsection*{{In terms of injective and projective resolutions}}\label{InTermsOfResolutions}

In the case that the underlying [[abelian category]] $\mathcal{A}$ has \href{injective%20object#EnoughInjectives}{enough injectives} or \href{projective%20object#EnoughInjectives}{enough projectives}, the [[hom sets]] in the derived category may equivalently be obtained as [[homotopy]]-classes of [[chain maps]] from [[projective resolutions]] to [[injective resolutions]] of chain complexes.

In view of the existence of the injective and projective [[model structure on chain complexes]] this is a special case of the general fact that [[homotopy categories]] of [[model categories]] may be obtained by forming homotopy classes of maps in the model category from [[cofibrant resolutions]] to [[fibrant resolutions]]. But here we spell out an direct discussion of this fact for chain complexes.

\begin{defn}
\label{}\hypertarget{}{}
Write $K^+(\mathcal{I}_{\mathcal{A}}) \hookrightarrow K^+(\mathcal{A})$ for the [[full subcategory]] of the [[homotopy category of chain complexes]] bounded above on those that are degreewise [[injective objects]].

Dually, let $K^-(\mathcal{P}_{\mathcal{A}}) \hookrightarrow K^-(\mathcal{A})$ for the [[full subcategory]] of the [[homotopy category of chain complexes]] bounded below on those that are degreewise [[projective objects]].

\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
If $\mathcal{A}$ has \href{http://ncatlab.org/nlab/show/injective%20object#EnoughInjectives}{enough injectives} then the canonical functor

\begin{displaymath}
K^+(\mathcal{I}_{\mathcal{A}}) \to D^+(\mathcal{A})
\end{displaymath}
is an [[equivalence of categories]].

Dually, if $\mathcal{A}$ has \href{http://ncatlab.org/nlab/show/projective%20object#EnoughProjectives}{enough projectives} then the canonical functor

\begin{displaymath}
K^-(\mathcal{P}_{\mathcal{A}}) \to D^-(\mathcal{A})
\end{displaymath}
is an [[equivalence of categories]].

\end{theorem}
For instance (\hyperlink{Schapira}{Schapira, cor. 7.3.2}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[derived functor]]
\item [[triangulated categories]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The original reference is the thesis of [[Verdier]]:

\begin{itemize}%
\item [[Verdier, Jean-Louis]], \emph{Des Cat\'e{}gories D\'e{}riv\'e{}es des Cat\'e{}gories Ab\'e{}liennes}, Ast\'e{}risque (Paris: Soci\'e{}t\'e{} Math\'e{}matique de France) 239. Available in \href{http://www.math.jussieu.fr/~maltsin/jlv.html}{electronic format} courtesy of [[Georges Maltsiniotis]].

\end{itemize}
A systematic discussion from the point of view of [[localization]] and [[homotopy theory]] is in section 13 of

\begin{itemize}%
\item [[Masaki Kashiwara]], [[Pierre Schapira]], \emph{[[Categories and Sheaves]]}

\end{itemize}
and, similarly, in section 7 of

\begin{itemize}%
\item [[Pierre Schapira]], \emph{Categories and homological algebra} (2011) (\href{http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf}{pdf})

\end{itemize}
A pedagogical introduction is

\begin{itemize}%
\item R. P. Thomas, \emph{Derived categories for the working mathematician} (\href{http://arxiv.org/abs/math.AG/0001045}{arXiv:math.AG/0001045})

\end{itemize}
A good survey of the more general topic of derived categories is

\begin{itemize}%
\item [[Bernhard Keller]], \emph{Derived categories and their uses} (\href{http://www.google.de/url?sa=t&source=web&ct=res&cd=6&ved=0CC8QFjAF&url=http%3A%2F%2Fwww.math.jussieu.fr%2F~keller%2Fpubl%2Fdcu.ps&rct=j&q=derived+category&ei=Ib76SsSdAsjb-QaAw7moDw&usg=AFQjCNGIgXLHlprAoR70bGLWQmyKGHDjTQ}{ps})

\end{itemize}
See in particular also the list of references given there.

Other lecture notes include

\begin{itemize}%
\item Theo B\"u{}hler, \emph{An introduction to the derived category} (\href{http://www.uni-math.gwdg.de/theo/intro-derived.pdf}{pdf})

\end{itemize}
For a discussion in the context of [[(∞,1)-category|(∞,1)-categories]] and in particular [[stable (∞,1)-category|stable (∞,1)-categories]] see \href{http://www.math.harvard.edu/~lurie/papers/DAG-I.pdf#page=53}{section 13, p. 53} of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Stable ∞-Categories]]}

\end{itemize}
For the applications of derived categories in [[algebraic geometry]], see

\begin{itemize}%
\item [[Dmitri Orlov]], \emph{Derived categories of coherent sheaves and equivalences between them}, Uspekhi Mat. Nauk, 2003, Vol. 58, issue 3(351), pp. 89--172, \href{http://www.mi.ras.ru/~orlov/papers/Uspekhi2003.pdf}{English translation (PDF)}


\item [[Aleksei Bondal]], [[Dmitri Orlov]], \emph{\href{http://www.mi.ras.ru/~orlov/papers/Compositio2001.pdf}{Reconstruction of a variety from the derived category and groups of autoequivalences}}, Compositio Mathematica 125 (03), 327-344. See also [[Bondal-Orlov reconstruction theorem]].


\item [[Daniel Huybrechts]], \emph{Fourier-Mukai Transforms in Algebraic Geometry}, Oxford University Press, USA, 2006.


\item [[Igor Dolgachev]], \href{http://www.math.lsa.umich.edu/~idolga/derived9.pdf}{\emph{Derived categories}}.


\item [[Andrei Caldararu]], \emph{Derived categories of coherent sheaves: a skimming}. Lecture notes from \emph{Algebraic Geometry: Presentations by Young Researchers} in Snowbird, Utah, July 2004. Available on \href{http://arxiv.org/abs/math/0501094}{arXiv}.



\end{itemize}
[[!redirects derived categories]] [[!redirects bounded derived category]] [[!redirects bounded derived categories]]

[[!redirects derived (infinity,1)-category]] [[!redirects derived (infinity,1)-categories]] [[!redirects derived dg-category]] [[!redirects derived dg-categories]]



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